I remember that a friend once asked me what I liked about mathematics, and I was ashamed about the quality of the answer I gave him. I don’t remember the answer, but I do remember the feeling of shame. One of the reasons I went to Ireland was that I wanted to get away from all that math, it had been too much, and I had been studying for too long against my will. It’s not that I never liked it, but the rigor of university life means that one must study at a pace that might be rather intense for one’s liking and tolerance levels.
I believe that among all the ways one can categorize mathematicians, a natural one is with two sets whose intersection is not empty. One set contains those mathematicians who do mathematics for its own sake, they care not how their theorems might relate to the world around them. The other set contains the more scientifically minded mathematicians who are interested in providing models for our physical reality.
As a student, I think I belonged to the intersection, but as time went on the theorems and the topics that we worked on were getting a bit too abstract, and I became more drawn to mathematics as a science rather than as an art. I believe that all mathematicians at some point belonged to the intersection of these two sets. Let me show you an example of a time of joy when I was a mathematician for its own sake. It was in real analysis class, and we proved the Stone-Weierstrass theorem that states that any continuous function can be approximated on an interval by polynomials to any degree of accuracy. Mathematicians love polynomials, because they are easy to work with. When I thought about a sequence of polynomials converging to the Weierstrass function on a random interval I felt a beautiful ecstasy, the ecstasy of mathematics.
Most of the time now I think of mathematics as a way to model the real world, but I think I will forever belong to the intersection.